Dualité des espaces de fonctions entières en dimension infinie

Dwyer III, Thomas A. W.

Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 151-195.

On étudie ici quelques espaces de fonctions holomorphes dans des domaines localement convexes, ayant comme cas particuliers les espaces de Fock holomorphes. Les espaces duaux sont caractérisés avec la transformation de Fourier-Borel pour des types d’holomorphie appropriés. On montre que ces espaces de fonctions sont de Fréchet-Schwartz (resp. de Silva, resp. nucléaires) quand leurs domaines sont des espaces de Silva (resp. de Fréchet-Schwartz, resp. nucléaires). Les conditions de croissance $p$-sommable des séries de Taylor qui y interviennent sont équivalentes aux conditions de croissance exponentielle employées par Martineau dans l’étude des équations différentielles d’ordre infini et sont plus maniables que celles-ci dans des questions de dualité en dimension infinie.

Dual pairs of spaces of holomorphic functions on locally convex domains are studied, yielding the holomorphic Fock spaces as particular cases. The duality is with respect to the Fourier-Borel transformation for appropriate holomorphy types. These functions spaces are shown to be Fréchet-Schwartz spaces (resp. Silva spaces, resp. nuclear). The $p$-summable growth conditions of Taylor series that intervene in their definitions are equivalent to the exponential growth conditions employed by Martineau in the study of differential equations of infinite order, and are more manageable in the treatment of duality questions in infinite dimension.

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